Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $288$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $8 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $6^{8}\cdot48^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48D8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.288.8.354 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}23&4\\16&13\end{bmatrix}$, $\begin{bmatrix}25&38\\16&37\end{bmatrix}$, $\begin{bmatrix}31&34\\16&5\end{bmatrix}$, $\begin{bmatrix}35&16\\40&1\end{bmatrix}$, $\begin{bmatrix}37&24\\0&5\end{bmatrix}$, $\begin{bmatrix}37&40\\16&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.8.j.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $4096$ |
Jacobian
| Conductor: | $2^{26}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{2}$ |
| Newforms: | 36.2.a.a$^{3}$, 72.2.d.b, 144.2.a.a, 288.2.d.b |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x t + y v + w u $ |
| $=$ | $x t - y v + z r + w u$ | |
| $=$ | $x u - y r - z t$ | |
| $=$ | $x u + y r - z t - w v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{11} - 4 x^{7} y^{4} + x^{3} y^{8} + 18 x^{2} y^{7} z^{2} + 108 x y^{6} z^{4} + 216 y^{5} z^{6} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:0:0:1)$, $(0:0:0:0:0:0:1:0)$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 24.72.4.ch.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
| $\displaystyle W$ | $=$ | $\displaystyle -z$ |
Equation of the image curve:
| $0$ | $=$ | $ 4Y^{2}+ZW $ |
| $=$ | $ X^{3}+YZ^{2}-YW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.144.8.j.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{11}-4X^{7}Y^{4}+X^{3}Y^{8}+18X^{2}Y^{7}Z^{2}+108XY^{6}Z^{4}+216Y^{5}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 48.144.4-48.ba.1.15 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.ba.1.50 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.bd.2.7 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.bd.2.58 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-24.ch.1.18 | $48$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.576.15-48.ca.2.30 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cb.2.10 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cc.2.13 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cd.2.10 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.ci.1.13 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cj.2.9 | $48$ | $2$ | $2$ | $15$ | $2$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cm.1.17 | $48$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.cn.2.3 | $48$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 48.576.17-48.k.2.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.l.1.14 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.m.1.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.n.2.10 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.o.1.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.p.2.9 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.q.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.r.2.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.s.1.22 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.t.2.13 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.u.1.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.v.2.2 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.w.2.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.x.1.8 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.y.1.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.z.2.10 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ba.1.4 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.bb.2.17 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.bc.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.bd.2.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.be.1.14 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.bf.2.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.bg.1.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.bh.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ij.1.42 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.jh.2.14 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ju.1.8 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.kc.2.27 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.lq.1.27 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.lr.1.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ls.1.13 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.lt.2.28 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |